What is Multiple Linear Regression?

Multiple linear regression is the most common form of linear regression analysis.  As a predictive analysis, the multiple linear regression is used to explain the relationship between one continuous dependent variable and two or more independent variables.  The independent variables can be continuous or categorical (dummy coded as appropriate).

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Example Questions Answered:

Do age and IQ scores effectively predict GPA?

Do weight, height, and age explain the variance in cholesterol levels?

Assumptions:

Regression residuals must be normally distributed.

A linear relationship is assumed between the dependent variable and the independent variables.

The residuals are homoscedastic and approximately rectangular-shaped.

Absence of multicollinearity is assumed in the model, meaning that the independent variables are not too highly correlated.

At the center of the multiple linear regression analysis is the task of fitting a single line through a scatter plot.  More specifically the multiple linear regression fits a line through a multi-dimensional space of data points.  The simplest form has one dependent and two independent variables.  The dependent variable may also be referred to as the outcome variable or regressand.  The independent variables may also be referred to as the predictor variables or regressors.

There are 3 major uses for multiple linear regression analysis.  First, it might be used to identify the strength of the effect that the independent variables have on a dependent variable.

Second, it can be used to forecast effects or impacts of changes.  That is, multiple linear regression analysis helps us to understand how much will the dependent variable change when we change the independent variables.  For instance, a multiple linear regression can tell you how much GPA is expected to increase (or decrease) for every one point increase (or decrease) in IQ.

Third, multiple linear regression analysis predicts trends and future values.  The multiple linear regression analysis can be used to get point estimates.  An example question may be “what will the price of gold be 6 month from now?”

When selecting the model for the multiple linear regression analysis, another important consideration is the model fit.  Adding independent variables to a multiple linear regression model will always increase the amount of explained variance in the dependent variable (typically expressed as R²).  Therefore, adding too many independent variables without any theoretical justification may result in an over-fit model.

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Related Pages:

Conduct and Interpret a Multiple Linear Regression

Assumptions of Multiple Linear Regression